Reading caretaker boss Brian McDermott is still looking for his first win in the Royals hotseat and he oversaw a desolate display, brightened only by Gylfi Sigurdsson's sixth goal of the season.
In the 11th minute Reading's Ivar Ingimarsson failed to completely clear a long pass forward, with Argyle's Jamie Mackie reacting to collect the loose ball.
The Pilgrims' top scorer showed an effective turn of pace and was brought down by Ingimarsson in the penalty area, leaving Judge to convert from 12 yards.
Reading probed to find a way back into a scrappy game but Mackie remained a threat and he should have doubled the advantage when he headed Chris Clark's cross narrowly wide in the 24th minute.
Judge then collected Gary Sawyer's pass before releasing Karl Duguid, whose dangerous ball across the Reading six-yard box was almost turned into his own net by Brynjar Gunnarson.
Five minutes before the interval, a last ditch tackle from Ryan Bertrand was required to thwart Mackie as the goal beckoned.
On the stroke of half-time, Argyle goalkeeper Romain Larrieu pushed a shot from Jobi McAnuff behind, following good link-up play involving Gunnarsson and Marek Matejovsky.
Argyle's Rory Fallon was able to break away from Reading's shaky backline 10 minutes into the second period.
The New Zealand international cut back to Clark whose shot grazed the outside of the upright.
Argyle doubled their lead in the 59th minute when Arnason rose unmarked to thump a header past Federici from Judge's corner.
But Reading hit back immediately.
Substitute Grzegorz Rasiak was instantly involved when his goalbound shot was scrambled out of the Pilgrims box.
McAnuff pumped the ball back into the danger area for Sigurdsson to head home.
The Royals' hopes was short lived, however, as Clark found space down the left wing and crossed for Judge to head his second after 63 minutes.
The visitors remained vulnerable at the back and Federici was alert to parry a rasping shot from Pilgrims substitute Ashley Barnes.
From the resulting corner it was Clark's turn to enjoy a free header, but he could not keep his effort down.
The misery was complete for McDermott's side when Mackie collected a quick free-kick to draw a sharp save from Federici, but Barnes was on hand to turn home the loose ball.